Perhaps it is just hard-wired into the minds of the human species
to make it a practice of overstating our case or perhaps it is
something we just learn as a strategy for impressing or persuading
others. Whatever the reason, we all are guilty of it from time to
time and we all should be wary of it in evaluating what others say.
But too often we are apt to ignore that cautionary advice.

An article I read recently claimed that by virtue of Arrow's
theorem we know that all voting methods are defective. Now while I
have little doubt that every possible voting method would be
defective in the opinion of someone, that is not exactly what Arrow's
theorem says. Arrow's theorem actually shows that ranked
voting systems are defective in a particular, well-defined way. A
ranked voting system (in the sense of Arrow's theorem) is one in
which each voter is asked to vote by specifying a ranking of all of
the candidates in order of preference. But that is a very special
kind of voting system and Arrow's theorem says nothing whatever
about many other voting systems.

A suspicious mind might wonder whether this overly broad
conclusion is an accident or whether it is purposely drawn by ranked voting advocates so as to suggest that we simply must be satisfied
with a demonstrably defective voting system. Such a person would
actually prefer that rather dismal conclusion to the alternative and
more natural realization that we might best turn to something other
than a ranked voting system. Fortunately there are many good
alternatives to ranked voting systems.

There is an understandable attraction to ranked voting that
derives from a mistaken impression that by ranking the candidates a
voter's opinions can fully, accurately and completely expressed.
This fallacy was addressed in the earliest of these
articles concerning ranked voting. The fact that the ranking may
overstate the feelings of a voter (who is forced to invent
preferences in ranking where none exist) seems just to be ignored;
we are just so accustomed to overstatement and tend to dismiss it as
being normal and harmless but it does disguise that the voter is in
some instances indifferent, something that needs to be reflected in
any complete and accurate measure of voter preference. But even
that is not the end of the story.

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Actual implementations of ranked voting generally relax its
definition to only require a voter to specify a ranking of just a few
of the candidates; this is no longer a ranked voting system. So
even in the relaxed real-world version of ranked voting, the
conclusion of Arrow's theorem possibly might not be justified.
Nonetheless, Arrow's theorem does suggest that if we want a voting
system without serious defects, we would best be advised to look
elsewhere than to ranked voting systems. But despite this evidence
from Arrow's theorem, there remains strong
support for the particular form of ranked voting called Instant-Runoff Voting (IRV), so the topic needs further attention. Is
the real-world version of IRV defective? An example could show it to
be.

Here we have seen three instances of overstatement -- in the
interpretation of Arrow's theorem as applying to all voting systems
(not just ranked ones), in the actual overstatement of preferences by
the voters, and finally in applying Arrow's theorem to voting systems
that allow voters not to rank all of the candidates. In everyday
parlance we may be accustomed to overstatement, but here we are
applying a mathematical theorem where precise adherence to details in
definitions and assumptions is critically important in this arena.
In mathematics, it is often the case that seemingly minor changes to
a definition or to assumptions can lead to very different
conclusions.

The subject of this
series of articles is balanced voting, and since none of the
balanced systems are ranked this is the first time that it has seemed
appropriate to even mention Arrow's theorem. Still, due to the
great popularity of IRV, that particular ranked system has been
discussed in several of the articles. One reason I wrote this
article was to have an opportunity to make available a brief survey
of these articles for someone who might be primarily interested in
IRV.

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In an
early article and again in a
more recent one, we addressed difficulties with the very rational
behind the invention of IRV. And in yet
another article we provided a concrete example to clearly
illustrate a serious defect in an IRV election. These articles
should make it clear that IRV falls well short of being the perfect
answer to better elections.

Finally turning to a slightly different topic, I should take this
opportunity to make it clear that Instant-Runoff Balanced Voting (IRBV) is a balanced voting system and not
a ranked system. At first glance, IRBV may seem like a ranked voting
system, but like other balanced systems, it is not actually a ranked
system and so Arrow's theorem does not say anything about IRBV.
However, IRBV is modeled on IRV and it shares much the same appeal as
IRV though it lacks many of its deficiencies. However, as with IRV,
our tradition of counting the votes in a distributed manner rather
than at a single central place is very problematic if not completely
impossible. More insight into the relationship between IRBV and IRV
is developed in one of the later
articles.